No problem for all the questions. That's what we're here for, and what your subscription entitles you to. Ask away!
When you say that you 'close everything down', this generally means that your session with Facebook will be closed too (logged out). In this case, if you return to the site you won't be automatically logged back in. If you're logged into Facebook in another tab, the 'Automatically log facebook users in' setting is enabled and then you're not seeing the login occur, let us know. That's not right behavior.
As for when you log in, it not redirecting you properly, it's probably your configuration for "Enable Returning User Redirection:". I would recommend setting this to 'no'. That way, when a use is logged in, they won't be directed to a different page on your site and should remain where they are. Hopefully that fixes that problem, but if not, let us know.
Regarding Graph functionality, more features will be added with every release. The next release (3.2, due out in a few weeks), will have the Open Graph API tags which let you specify things like the Image and Link to use when a user Like's a page. However, it's important to note that not all features will be implemented directly by us. The link you point to is a way to 'get' information from Facebook. You can already do a lot of Graph calls with JFBConnect right now, but the display, formatting, and use of that data would be very site dependent. If you have (or are) a developer, we can gladly help you to implement features to fetch this information, but we won't be putting a default view for all possible Graph calls for quite a while, if ever. Hope that makes sense as to why.
Finally, regarding LiveStream, in 3.1, there's no tag for adding that to your site. In 3.2, that tag (as well as all the other module tags) will be available. For now, to include it in an article, you should create an instance of the JFBCLiveFeed module, put it in the position of "jfbclivefeed" (for example). Then, in your article, you can just put {loadposition jfbclivefeed}. This uses the Content - Load Module plugin, included with Joomla, to load a specific module. That should do what you're looking for.
Good luck, and let us know if you have any other questions!